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The derivative is "better division", where you get the speed through the continuum at every instant. When your speed changes as you go, you need to describe your speed at each instant. That's the derivative. If you apply this changing speed to each instant take the integral of the derivative , you recreate the original behavior, just like applying the daily stock market changes to recreate the full price history. But this is a big topic for another day. General formula for change. We found the "perfect" model by making a measurement and improving it.

These are called "discontinuous" functions, which is essentially "cannot be modeled with limits". As you can guess, the derivative doesn't work on them because we can't actually predict their behavior. Discontinuous functions are rare in practice, and often exist as "Gotcha!

Realize the theoretical limitation of derivatives, and then realize their practical use in measuring every natural phenomena. Nearly every function you'll see sine, cosine, e, polynomials, etc. The relationship between derivatives, integrals and anti-derivatives is nuanced and I got it wrong originally. Here's a metaphor. Start with a plate, your function to examine:.

There's no algorithm to find the anti-derivative; we have to guess. Finding derivatives is mechanics; finding anti-derivatives is an art. Sometimes we get stuck: we take the changes, apply them piece by piece, and mechanically reconstruct a pattern. It might not be the "real" original plate, but is good enough to work with.

Another subtlety: aren't the integral and anti-derivative the same? That's what I originally thought. Yes, but this isn't obvious: it's the fundamental theorem of calculus! Yes, but this isn't obvious: it's the Pythagorean theorem! Thanks to Joshua Zucker for helping sort me out. Math is a language, and I want to "read" calculus not "recite" calculus, i. I need the message behind the definitions.

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BetterExplained helps k monthly readers with friendly, insightful math lessons more. Calculus: Building Intuition for the Derivative. How do you wish the derivative was explained to you? Here's my take. The derivative is the heart of calculus, buried inside this definition: But what does it mean? Derivatives create a perfect model of change from an imperfect guess.

We all live in a shiny continuum Infinity is a constant source of paradoxes "headaches" : A line is made up of points? So there's an infinite number of points on a line?

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How do you cross a room when there's an infinite number of points to visit? Gee, thanks Zeno. Analogy: See division as a rate of motion through a continuum of points What's after zero? Another brain-buster: What number comes after zero? Anything you can name, I can name smaller I'll just halve your number I don't know exactly how big it is, but it's there! Analogy: dx is a "jump" to the next number in the continuum. Measurements depend on the instrument The derivative predicts change.

1. Limits and Differentiation

Ok, how do we measure speed change in distance? Officer: Do you know how fast you were going? Driver: I have no idea.

Officer: 95 miles per hour. Driver: But I haven't been driving for an hour!

Find the derivative of the function and evaluate the derivative at the given value of a.

Running the Treadmill We're nearing the chewy, slightly tangy center of the derivative. What to do? Analogy: Remove the "electrode effect" after making your measurement By the way, the "electrode effect" shows up everywhere. The absolute change between each result is: 1, 3, 5, 7, 9, 11, 13, What is "5" made of? The derivative as "continuous division" I see the integral as better multiplication , where you can apply a changing quantity to another. General formula for change "The derivative is 44" means "At our current location, our rate of change is Gotcha: Our models may not be perfect We found the "perfect" model by making a measurement and improving it.

Gotcha: Integration doesn't really exist The relationship between derivatives, integrals and anti-derivatives is nuanced and I got it wrong originally. Start with a plate, your function to examine: Differentiation is breaking the plate into shards. Integration is weighing the shards: your original function was "this" big.

There's a procedure, cumulative addition, but it doesn't tell you what the plate looked like. Anti-differentiation is figuring out the original shape of the plate from the pile of shards. The result is. To differentiate f with respect to the variable s , enter. Basically, the default variable is the letter closest to x in the alphabet. See the complete set of rules in Find a Default Symbolic Variable.

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In the preceding example, diff f takes the derivative of f with respect to t because the letter t is closer to x in the alphabet than the letter s is. Calculate the second derivative of f with respect to t :. Note that diff f, 2 returns the same answer because t is the default variable. To differentiate the Bessel function of the first kind, besselj nu,z , with respect to z , type.

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The diff function can also take a symbolic matrix as its input. In this case, the differentiation is done element-by-element. Consider the example. You can also perform differentiation of a vector function with respect to a vector argument. To calculate the Jacobian matrix, J , of this transformation, use the jacobian function. The mathematical notation for J is.

The commands. The arguments of the jacobian function can be column or row vectors.

Finding a Derivative Using the Definition of a Derivative

Moreover, since the determinant of the Jacobian is a rather complicated trigonometric expression, you can use simplify to make trigonometric substitutions and reductions simplifications. A table summarizing diff and jacobian follows.